from remez import *
from visualization import *
import pandas as pd

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

fx = "np.arctan(np.sqrt(3 + x**3) - np.exp(1 + x))"
g = "(np.sqrt(3 + x**3) - np.exp(1 + x))"
g_prime = "((3 * x**2) / (2 * np.sqrt(3 + x**3)) - np.exp(1 + x))"
fx_der = f"({g_prime}) / (1 + {g}**2)"
interval = [np.sqrt(2), np.pi**2]
n = 5
px, xn, history = remez(fx, fx_der, interval, n)
visualization(fx, px, xn, history, interval, n)

f(x) = np.arctan(np.sqrt(3 + x**3) - np.exp(1 + x)), interval = [1.4142135623730951, 9.869604401089358]

Polynomial degree = 5
Pn(x):
- 3.955756933047265e-05 * x**5 + 0.0012947712130833584 * x**4 - 0.01654139703555944 * x**3 + 0.10351664953941357 * x**2 - 0.32051562487328494 * x - 1.1703528319321932

xn points:
[1.41421356 1.83693284 3.14845216 5.17561358 7.42752857 9.19850669
 9.8696044 ]

Converge iteration: 7
MAE of approximation: 0.0012079008992569307
MAE of interpolation: 0.0021889162615584823

fx = "np.sin(x)"
fx_der = "np.cos(x)"
interval = [-10, 10]
n = 10
px, xn, history = remez(fx, fx_der, interval, n)
visualization(fx, px, xn, history, interval, n)

f(x) = np.sin(x), interval = [-10, 10]

Polynomial degree = 10
Pn(x):
6.407420434363546e-23 * x**10 + 1.6453249038659575e-07 * x**9 - 2.0501296218005414e-20 * x**8 - 3.9764054890758586e-05 * x**7 + 1.5820289608296249e-18 * x**6 + 0.0031818236931100714 * x**5 - 1.0025210470445e-17 * x**4 - 0.0922407606343639 * x**3 - 8.904546096582622e-16 * x**2 + 0.6882375433917891 * x + 1.7319479184152442e-14

xn points:
[-10.          -9.52836569  -8.19876035  -6.22807034  -3.86512566
  -1.30745184   1.30745184   3.86512566   6.22807034   8.19876035
   9.52836569  10.        ]

Converge iteration: 13
MAE of approximation: 0.2599467004604461
MAE of interpolation: 0.37372945033890925

fx = "np.abs(np.sin(x))"
fx_der = "np.sign(np.sin(x)) * np.cos(x)"
interval = [0, 3]
n = 10
px, xn, history = remez(fx, fx_der, interval, n)
visualization(fx, px, xn, history, interval, n)

f(x) = np.abs(np.sin(x)), interval = [0, 3]

Polynomial degree = 10
Pn(x):
- 2.6041641511055603e-07 * x**10 + 4.090815781454296e-06 * x**9 - 4.157868757727268e-06 * x**8 - 0.00019000590615703394 * x**7 - 1.124543122012178e-05 * x**6 + 0.008343191246311606 * x**5 - 5.4638087840026595e-06 * x**4 - 0.16666488277893782 * x**3 - 3.0059822135403053e-07 * x**2 + 1.0000000198265397 * x - 2.1640269330467054e-10

xn points:
[0.         0.05354876 0.21047198 0.45989084 0.78460359 1.16233502
 1.56726832 1.97167627 2.34749089 2.66726756 2.90189731 3.        ]

Converge iteration: 7
MAE of approximation: 2.1640367275921335e-10
MAE of interpolation: 3.936732029553218e-10

fx1 = "np.sin(x)"
fx2 = "-np.sin(x)"
fx_der1 = "np.cos(x)"
fx_der2 = "-np.cos(x)"
interval = [-10, 10]
n = 10

px1, xn1, history1 = remez(fx1, fx_der1, interval, n)
px2, xn2, history2 = remez(fx2, fx_der2, interval, n)
# xn1 == xn2
# history1 == history2

fx = "np.abs(np.sin(x))"
px = f"np.piecewise(x, [np.sin(x) >= 0, (np.sin(x) < 0)], [{px1}, {px2}])"
visualization(fx, px, xn1, history1, interval, n, compare_interpolation=True)

f(x) = np.abs(np.sin(x)), interval = [-10, 10]

Polynomial degree = 10
Pn(x):
np.piecewise(x, [np.sin(x) >= 0, (np.sin(x) < 0)], [6.407420434363546e-23 * x**10 + 1.6453249038659575e-07 * x**9 - 2.0501296218005414e-20 * x**8 - 3.9764054890758586e-05 * x**7 + 1.5820289608296249e-18 * x**6 + 0.0031818236931100714 * x**5 - 1.0025210470445e-17 * x**4 - 0.0922407606343639 * x**3 - 8.904546096582622e-16 * x**2 + 0.6882375433917891 * x + 1.7319479184152442e-14, - 6.407420434363546e-23 * x**10 - 1.6453249038659575e-07 * x**9 + 2.0501296218005414e-20 * x**8 + 3.9764054890758586e-05 * x**7 - 1.5820289608296249e-18 * x**6 - 0.0031818236931100714 * x**5 + 1.0025210470445e-17 * x**4 + 0.0922407606343639 * x**3 + 8.904546096582622e-16 * x**2 - 0.6882375433917891 * x - 1.7319479184152442e-14])

xn points:
[-10.          -9.52836569  -8.19876035  -6.22807034  -3.86512566
  -1.30745184   1.30745184   3.86512566   6.22807034   8.19876035
   9.52836569  10.        ]

Converge iteration: 13
MAE of approximation: 0.2599467004604461
MAE of interpolation: 0.9198614658691844

fx = "np.piecewise(x, [x < 0, (x >= 0) & (x <= 2), x > 2], [lambda x: x**2, lambda x: x + 1, np.sin])"
fx_der = "np.piecewise(x, [x < 0, (x >= 0) & (x <= 2), x > 2], [lambda x: 2 * x, 1, np.cos])"
interval = [-3, 5]
n = 15
px, xn, history = remez(fx, fx_der, interval, n)
visualization(fx, px, xn, history, interval, n)

f(x) = np.piecewise(x, [x < 0, (x >= 0) & (x <= 2), x > 2], [lambda x: x**2, lambda x: x + 1, np.sin]), interval = [-3, 5]

Polynomial degree = 15
Pn(x):
- 3.0476781469839864e-09 * x**15 - 2.8555206317755127e-07 * x**14 + 4.591585509723426e-06 * x**13 - 9.167078961250668e-06 * x**12 - 0.00015078529781879918 * x**11 + 0.0005725252066228006 * x**10 + 0.002132562337285741 * x**9 - 0.010045947899489453 * x**8 - 0.018589448071827566 * x**7 + 0.0872691674876504 * x**6 + 0.12659221011160046 * x**5 - 0.40150792018637294 * x**4 - 0.7592763866710046 * x**3 + 1.2497608678289127 * x**2 + 1.1414305084966663 * x + 0.5669558632090834

xn points:
[-3.         -2.92448242 -2.70480454 -2.36242476 -1.93657102 -1.49184043
 -1.13051067 -0.98559781  0.90073722  2.9677204   3.11650621  3.48492005
  3.93466818  4.3626498   4.70533835  4.92468226  5.        ]

Converge iteration: 50
MAE of approximation: 1.5634692716080894
MAE of interpolation: 1.8112270478784378

# set some hyper-parameter
interval = [-1, 1]
degree = 40
half = degree // 2
func_dict = {
    r"$e^x$": [("np.exp(x)", "np.exp(x)", 2 * i + 1) for i in range(half)],
    r"$e^{1+x^2}$": [("np.exp(1 + x**2)", "2*x*np.exp(1+x**2)", 2 * i + 1) for i in range(half)],
    r"$\ln{(1+x^2)}$": [("np.log(1 + x**2)", "2*x/(1+x**2)", 2 * i + 1) for i in range(half)],
    r"$\sqrt{1+x^2}$": [("np.sqrt(1 + x**2)", "x/np.sqrt(1+x**2)", 2 * i + 1) for i in range(half)],
    r"$\frac{1}{1+x^2}$": [
        ("(1 + x**2)**-1", "-2 * x * (1 + x**2)**-2", 2 * i + 1) for i in range(half)
    ],
}

df = pd.DataFrame([2 * i + 1 for i in range(half)], columns=["n"])
df.set_index(keys="n", inplace=True)

for func_name, function in func_dict.items():
    store = []
    history_error = np.inf
    for fx, fx_der, n in function:
        px, xn, history = remez(fx, fx_der, interval, n)

        max_interval_error = visualization_pipeline(fx, px, xn, history, interval, n, history_error)

        if max_interval_error is not None:
            store.append((n, max_interval_error))
            history_error = max_interval_error
        else:
            y = pd.DataFrame(
                [j[1] for j in store], columns=[func_name], index=[j[0] for j in store]
            )
            df = pd.merge(df, y, left_index=True, right_index=True, how="left")
            break

f(x) = np.exp(x), interval = [-1, 1]

Polynomial degree = 1
Pn(x):
1.1752011936438014 * x + 1.2642790490197413

xn points:
[-1.          0.16143936  1.        ]

Converge iteration: 2
MAE of approximation: 0.2788015857955024
MAE of interpolation: 0.3722483506650818

f(x) = np.exp(x), interval = [-1, 1]

Polynomial degree = 3
Pn(x):
0.17953348361616284 * x**3 + 0.5429727883818615 * x**2 + 0.9956677100276385 * x + 0.9945794763246946

xn points:
[-1.         -0.68223286  0.04954372  0.73170678  1.        ]

Converge iteration: 10
MAE of approximation: 0.005528370108687675
MAE of interpolation: 0.006656866235436709

f(x) = np.exp(x), interval = [-1, 1]

Polynomial degree = 5
Pn(x):
0.008738191001535804 * x**5 + 0.04379369637407595 * x**4 + 0.1664246561337558 * x**3 + 0.49919698263496903 * x**2 + 1.0000383465085099 * x + 1.0000447502942726

xn points:
[-1.         -0.860197   -0.48239233  0.02369355  0.51792608  0.8720372
  1.        ]

Converge iteration: 5
MAE of approximation: 4.5205511926216246e-05
MAE of interpolation: 5.179584768422174e-05

f(x) = np.exp(x), interval = [-1, 1]

Polynomial degree = 7
Pn(x):
0.00020540802649050407 * x**7 + 0.0014392723790619436 * x**6 + 0.008327356909716574 * x**5 + 0.04163501496769684 * x**4 + 0.16666860522273672 * x**3 + 0.5000063462450856 * x**2 + 0.9999998234848574 * x + 0.9999998013981222

xn points:
[-1.         -0.92189579 -0.70029889 -0.37097185  0.01384577  0.39460855
  0.71414566  0.92595162  1.        ]

Converge iteration: 11
MAE of approximation: 1.9982527765805003e-07
MAE of interpolation: 2.224393229610655e-07

f(x) = np.exp(x), interval = [-1, 1]

Polynomial degree = 9
Pn(x):
2.831819307838153e-06 * x**9 + 2.549918543906719e-05 * x**8 + 0.00019832833996316233 * x**7 + 0.001388275963657203 * x**6 + 0.008333374802244744 * x**5 + 0.041666886016745786 * x**4 + 0.16666665818280604 * x**3 + 0.49999997254821943 * x**2 + 1.0000000004994796 * x + 1.0000000005494574

xn points:
[-1.         -0.95020991 -0.80594349 -0.5819317  -0.3008731   0.00907121
  0.31728375  0.59380785  0.81221331  0.95194303  1.        ]

Converge iteration: 20
MAE of approximation: 5.517248879982617e-10
MAE of interpolation: 6.027107701811474e-10

f(x) = np.exp(x), interval = [-1, 1]

Polynomial degree = 11
Pn(x):
2.561995535192282e-08 * x**11 + 2.8190179028409166e-07 * x**10 + 2.754958672734437e-06 * x**9 + 2.4794443049258586e-05 * x**8 + 0.00019841320798506375 * x**7 + 0.0013888925996554246 * x**6 + 0.008333333168188317 * x**5 + 0.04166666579607343 * x**4 + 0.1666666666899577 * x**3 + 0.5000000000746724 * x**2 + 0.9999999999990423 * x + 0.9999999999989624

xn points:
[-1.         -0.96550556 -0.86446038 -0.70395441 -0.49526455 -0.25288331
  0.00639971  0.26482165  0.50486985  0.71035491  0.86766177  0.96636289
  1.        ]

Converge iteration: 8
MAE of approximation: 1.0407230632836217e-12
MAE of interpolation: 1.120881165661558e-12

f(x) = np.exp(1 + x**2), interval = [-1, 1]

Polynomial degree = 1
Pn(x):
0.0 * x + 5.0536689636948475

xn points:
[-1.000000e+00  6.123234e-17  1.000000e+00]

Converge iteration: 1
MAE of approximation: 2.335387135235803
MAE of interpolation: 2.907367028592586

f(x) = np.exp(1 + x**2), interval = [-1, 1]

Polynomial degree = 3
Pn(x):
0.0 * x**3 + 4.670774270471606 * x**2 + 2.4303249480189244

xn points:
[-1.00000000e+00 -7.35747822e-01 -2.31144126e-54  7.35747822e-01
  1.00000000e+00]

Converge iteration: 2
MAE of approximation: 0.28795688044012024
MAE of interpolation: 0.3365100679975299

f(x) = np.exp(1 + x**2), interval = [-1, 1]

Polynomial degree = 5
Pn(x):
- 4.196614930524684e-15 * x**5 + 2.2997403935142002 * x**4 + 5.285740251788736e-15 * x**3 + 2.3234312053486357 * x**2 - 1.0891253212640522e-15 * x + 2.742083164263429

xn points:
[-1.00000000e+00 -8.74855034e-01 -5.15516668e-01  4.23832297e-17
  5.15516668e-01  8.74855034e-01  1.00000000e+00]

Converge iteration: 5
MAE of approximation: 0.023801335804385815
MAE of interpolation: 0.026817341406789375

f(x) = np.exp(1 + x**2), interval = [-1, 1]

Polynomial degree = 7
Pn(x):
- 8.531482125457164e-15 * x**7 + 0.761055002578163 * x**6 + 1.7371043166070163e-14 * x**5 + 1.1463076373076815 * x**4 - 1.0941057546560405e-14 * x**3 + 2.763411630585761 * x**2 + 2.101496505947406e-15 * x + 2.716800931428885

xn points:
[-1.00000000e+00 -9.27237104e-01 -7.15870878e-01 -3.90765952e-01
 -5.44905272e-14  3.90765952e-01  7.15870878e-01  9.27237104e-01
  1.00000000e+00]

Converge iteration: 6
MAE of approximation: 0.001480897030161188
MAE of interpolation: 0.0016309882409331067

df.dropna(how="all", inplace=True)
df

x_length = df.index[-1]
df.plot(
    figsize=(12, 6),
    title=f"log-Max-Error Versus Degree with Interval {interval}",
    logy=True,
    style=["rd-", "d--", "d-", "d-", "d-"],
    xlim=(0.5, x_length + 0.5),
    ylim=[5 * 10**-16, 1],
    xticks=np.arange(1, x_length + 1, 2),
    yticks=[10**-i for i in range(16)],
    fontsize=12,
    xlabel="Polynomial Degree",
    ylabel="log-Max-Error",
)
plt.savefig("./images/log-Max-Error_Versus_Degree.png", dpi=300)
plt.show()

	$e^x$	$e^{1+x^2}$	$\ln{(1+x^2)}$	$\sqrt{1+x^2}$	$\frac{1}{1+x^2}$
n
1	2.788016e-01	2.335387e+00	3.465736e-01	2.071068e-01	2.499747e-01
3	5.528370e-03	2.879569e-01	2.982624e-02	8.882790e-03	4.289322e-02
5	4.520551e-05	2.380134e-02	3.423981e-03	7.638368e-04	7.358156e-03
7	1.998253e-07	1.480897e-03	4.415460e-04	8.197651e-05	1.262658e-03
9	5.517249e-10	7.383511e-05	6.071410e-05	9.869364e-06	2.165439e-04
11	1.040723e-12	3.070489e-06	8.691045e-06	1.271337e-06	3.716919e-05
13	NaN	1.095056e-07	1.279333e-06	1.715430e-07	6.374165e-06
15	NaN	3.418386e-09	1.921678e-07	2.392230e-08	1.094159e-06
17	NaN	9.487788e-11	2.933012e-08	3.423567e-09	1.877271e-07
19	NaN	NaN	4.520698e-09	4.988316e-10	3.220903e-08
21	NaN	NaN	7.070591e-10	7.405032e-11	5.525997e-09
23	NaN	NaN	1.112376e-10	1.111800e-11	9.481455e-10
25	NaN	NaN	1.762368e-11	1.688649e-12	1.620220e-10
27	NaN	NaN	2.807810e-12	2.584599e-13	2.791084e-11
29	NaN	NaN	4.499734e-13	4.041212e-14	4.787837e-12
31	NaN	NaN	7.235185e-14	6.217249e-15	8.217871e-13
33	NaN	NaN	NaN	1.110223e-15	1.411093e-13
35	NaN	NaN	NaN	NaN	2.398082e-14
37	NaN	NaN	NaN	NaN	4.662937e-15

Examples¶

Single function usage¶

Pipeline¶

Statistics¶